Question

please tell the shortest derivation for newtons law of cooling

Answer 1

Other differential equations

We have examined the behaviour of two simple differential equations so far, one for population growth, and one for the radioactive decay of a substance. The methods we have developed are actually useful for many other interesting problems, and can help us to make predictions about other systems that, at first sight, do not seem at all related. We will find that the common thread in all these systems is the simple differential equation of the form
This equation is of interest for either positive or negative values of the constant In fact, in the examples studied so far, we looked at one case in which and another case in which
Before continuing, let us recall that the behaviour of the solution(s) to this equation depend on whether the constant  is positive or negative:

Answer 2

Newton's law of cooling states that the rate of change of the temperature of an object is proportional to the difference between its own temperature and the ambient temperature (i.e. the temperature of its surroundings).

Newton's Law makes a statement about an instantaneous rate of change of the temperature.

We will see that when we translate this verbal statement into a differential equation, we arrive at a differential equation.

The solution to this equation will then be a function that tracks the complete record of the temperature over time.